\section{\label{sec:setup}Experimental Setup and Data Taking}

The setup of the OSQAR photon regeneration experiment is schematically shown in Figure \ref{fig:lsw}. As in all LSW experiments, the operation principle is based on the possible generation of an axion-like particle from a polarized laser beam in the background of a magnetic field via the di-photon vertex and the process $\gamma\gamma \rightarrow a$. Once an ALP is created, it will propagate along the same direction as the initial laser photon, transverse the wall due its low coupling strength to matter and enter the \textit{regeneration area} on the second side of the experiment, while all photons are blocked by the wall in between. 

\begin{figure}[ht]
    \centering
    \includegraphics[width=0.45\textwidth]{Figures/lsw.png}
    \caption{Scheme of the operation principle of a general LSW experiment.}
\label{fig:lsw}
\end{figure}

A second external magnetic field is applied within the regeneration area, allowing the ALP to reconvert into a photon via the opposite process $a\,\gamma\rightarrow \gamma$, that can be subsequently be detected by a photon sensor.

The di-photon coupling of scalar- and pseudo-scalar ALPs can be expressed in the Lagrangian formulation as 

\begin{equation}
    \mathcal{L}_{Int} = \frac{1}{4} g_a F_{\mu\nu}\widetilde{F}^{\mu\nu}
                      = g_a \,(\mathbf{E}^2 - \mathbf{B}^2)
    \label{eq:lagrangian1}
\end{equation}

and

\begin{equation}
    \mathcal{L}_{Int} = \frac{1}{4} g_a F_{\mu\nu}\widetilde{F}^{\mu\nu}
                      = g_a \,(\mathbf{E} \cdot \mathbf{B}),
    \label{eq:lagrangian2}
\end{equation}

respectively, where $g_a$ is the corresponding coupling constant.

While the direction of the magnetic field $\mathbf{B}$ must be perpendicular to the laser polarization direction for scalar ALPs (Equation \ref{eq:lagrangian1}), the direction of  $\mathbf{B}$ must be parallel to the polarization direction of photons for pseudo-scalar ALPs search (Equation \ref{eq:lagrangian2}). In both cases the probability of the process $\gamma\gamma \rightarrow a$ and $a \rightarrow \gamma\gamma$ is given by~\cite{VanBibber:1987rq}:
\begin{equation}
    P_{\gamma\leftrightarrow a} = \frac{1}{4} {(\gagg BL)}^2
        {\left(
            \frac{2}{qL} \sin \frac{qL}{2}
        \right)}^2
    \label{eq:probability}
\end{equation}

Here\footnote{Units are in Heaviside-Lorentz system $(\hbar=c=1)$.} $\gagg$ denotes the ALPs di-photon coupling strength, while $q = | \omega - k_W |$ represents the momentum transfer, where $\omega$ is the energy of the photon and $k_W = {(\omega^2 - m_W^2)}^{1/2}$ is the momentum of the WISP of mass $m_W$.

Since the conversion probabilities for the WISP generation and the photon regeneration process are the same, $P_{\gamma\leftrightarrow W}=P_{W\leftrightarrow \gamma}$, the flux of detected, reconverted photons is given by 
\begin{equation}
    \frac{dN}{dt} = \frac{P}{\omega} \eta \; {(P_{\gamma\leftrightarrow W})}^2,
    \label{eq:flux}
\end{equation}

where $\eta$ denotes the photon detection efficiency and $P$ the optical power. It should be noted, that the expected photon rate, predicted by Equation \ref{eq:flux}, scales with the length and the strength of the magnetic field as the factor ${(BL)}^4$. Hence the use of strong magnetic fields over a wide range is the driving factor for the sensitivity of the experimental setup. Therefore OSQAR uses one LHC dipole on the generation and regeneration side, respectively. Both magnets are cooled down to 1.9\,K with superfluid He and provide a uniform transverse magnetic field with a strength of 9.5\,T over a magnetic length of 14.3\,m.  

We used an all-solid-state, single-frequency laser (COHERENT Verdi V18) with a wavelength of 532\,nm and a power of 18.5 W as high intensity photon source during data-taking. This corresponds to photons with an energy of 2.33\,eV per second. A beam expander telescope has been used to reduce the laser beam divergence. Since the optical beam is linearly polarized with a vertical orientation, a $\lambda/2$ wave-plate was inserted between the laser and the first LHC dipole in order to align the polarization of the light in the horizontal direction. This introduces an optical power loss of 20\% at the laser wavelengths. The beam profile of the laser was measured at several locations at the experiment and can be described by a gaussian distribution. 

The laser light after the second magnet was focused by an optical lens on a thermoelectric cooled CCD camera with an AR coated window (ANDOR DU934P-BEX2-DD). The camera was cooled down to $-95^\circ$C to ensure an efficient suppression of dark currents during operation. The CCD chip has an active area of $13.3\times 13.3\  mm^2$ and comprises a 2D array of $1024 \times 1024$ square pixels with a width of $13\ \mu m$. 90\% of the laser beam have been focussed on an area of $\approx0.8\,\mbox{mm}^2$, i.e. covering not more than 4 pixels. The photon detection efficiency of the whole setup, including CCD quantum efficiency and reflection losses in the optical setup, was measured with \SI{0.56(2)}{ADU/photon} at the given wavelength. The recorded photon distribution on the CCD camera, called \textit{frame} in the following, is variable in its length. 

The data taking was performed in August 2014 and comprises in total 119 runs. The measurement procedure for each run consist of three consecutive steps. As a first step, the beam position has to be measured. For this, the laser power is reduced to 3\,W and additional attenuators are installed. The light barrier between the two magnets is then removed. We recorded three frames for the beam position measurement with an exposure time of 0.01\,s and a 60\,s pause between the end of each frame and the start of a next one. This allows us to test and study the high frequency movements of the laser during the data taking. After the beam position measurements are performed, the light barrier is closed, the attenuators are removed and the laser power is set to 18.5\,W. As a second step, the data taking for the ALPs search is initiated. Two frames with an exposure time of 5400\,s are taken with a pause of 100\,s in between. A longer recording times for a single frame would lead to problems with the removal procedure of cosmic ray signatures. In a third step, the laser beam position measurement, analogous to the first step, is repeated.

In total, 60 runs with an applied magnetic field parallel to the laser polarization, and 59 runs with an applied magnetic field perpendicular to the laser polarization have been taken. Of these, 48 and 41 runs for the parallel and perpendicular cases have been used for further analysis, respectively.  In the remaining runs, the magnetic field was not stable, or temperature problems of the CCD camera occurred. In two of the runs, a cosmic ray muon was detected close to our predefined signal region. 
